LMFDB - Embedded newform 4020.2.g.c.1609.35

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1609,4020)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4020.1609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.35
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.c.1609.16

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(1.96624 - 1.06486i) q^{5} -4.32230i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(1.96624 - 1.06486i) q^{5} -4.32230i q^{7} -1.00000 q^{9} +3.68751 q^{11} +4.58547i q^{13} +(1.06486 + 1.96624i) q^{15} +7.89141i q^{17} +4.22571 q^{19} +4.32230 q^{21} +1.04462i q^{23} +(2.73216 - 4.18752i) q^{25} -1.00000i q^{27} -5.93796 q^{29} +5.66037 q^{31} +3.68751i q^{33} +(-4.60263 - 8.49866i) q^{35} +11.2388i q^{37} -4.58547 q^{39} +1.36096 q^{41} +6.90749i q^{43} +(-1.96624 + 1.06486i) q^{45} +0.227862i q^{47} -11.6823 q^{49} -7.89141 q^{51} -4.37470i q^{53} +(7.25051 - 3.92667i) q^{55} +4.22571i q^{57} +6.83840 q^{59} +4.41474 q^{61} +4.32230i q^{63} +(4.88286 + 9.01611i) q^{65} +1.00000i q^{67} -1.04462 q^{69} -6.97099 q^{71} -1.49940i q^{73} +(4.18752 + 2.73216i) q^{75} -15.9385i q^{77} -5.16488 q^{79} +1.00000 q^{81} +10.3758i q^{83} +(8.40322 + 15.5164i) q^{85} -5.93796i q^{87} +2.71428 q^{89} +19.8198 q^{91} +5.66037i q^{93} +(8.30873 - 4.49977i) q^{95} -8.15818i q^{97} -3.68751 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) $ 38 * q - 2 * q^5 - 38 * q^9	$ 38 q - 2 q^{5} - 38 q^{9} + 24 q^{11} + 2 q^{15} - 16 q^{19} + 12 q^{21} + 4 q^{25} - 56 q^{29} - 4 q^{31} - 2 q^{35} + 60 q^{41} + 2 q^{45} - 70 q^{49} + 12 q^{55} - 52 q^{59} + 48 q^{61} + 16 q^{65} - 12 q^{69} + 12 q^{75} - 24 q^{79} + 38 q^{81} + 16 q^{85} - 76 q^{89} + 44 q^{91} + 36 q^{95} - 24 q^{99}+O(q^{100}) $ 38 * q - 2 * q^5 - 38 * q^9 + 24 * q^11 + 2 * q^15 - 16 * q^19 + 12 * q^21 + 4 * q^25 - 56 * q^29 - 4 * q^31 - 2 * q^35 + 60 * q^41 + 2 * q^45 - 70 * q^49 + 12 * q^55 - 52 * q^59 + 48 * q^61 + 16 * q^65 - 12 * q^69 + 12 * q^75 - 24 * q^79 + 38 * q^81 + 16 * q^85 - 76 * q^89 + 44 * q^91 + 36 * q^95 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$.

$n$	$1141$	$2011$	$2681$	$3217$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	1.96624	−	1.06486i	0.879327	−	0.476218i
$5$	1.96624	−	1.06486i	0.879327	−	0.476218i
$6$		0			0
$7$		−	4.32230i		−	1.63368i	−0.576867	−	0.816838i	$-0.695725\pi$
$7$		−	4.32230i		−	1.63368i	0.576867	−	0.816838i	$-0.304275\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	3.68751			1.11183			0.555913	−	0.831240i	$-0.312369\pi$
$11$	3.68751			1.11183			0.555913	+	0.831240i	$0.312369\pi$
$12$		0			0
$13$			4.58547i			1.27178i	0.771780	+	0.635890i	$0.219367\pi$
$13$			4.58547i			1.27178i	−0.771780	+	0.635890i	$0.780633\pi$
$14$		0			0
$15$	1.06486	+	1.96624i	0.274945	+	0.507680i
$16$		0			0
$17$			7.89141i			1.91395i	0.290172	+	0.956974i	$0.406287\pi$
$17$			7.89141i			1.91395i	−0.290172	+	0.956974i	$0.593713\pi$
$18$		0			0
$19$	4.22571			0.969444			0.484722	−	0.874668i	$-0.338921\pi$
$19$	4.22571			0.969444			0.484722	+	0.874668i	$0.338921\pi$
$20$		0			0
$21$	4.32230			0.943203
$22$		0			0
$23$			1.04462i			0.217818i	0.994052	+	0.108909i	$0.0347357\pi$
$23$			1.04462i			0.217818i	−0.994052	+	0.108909i	$0.965264\pi$
$24$		0			0
$25$	2.73216	−	4.18752i	0.546433	−	0.837503i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−5.93796			−1.10265			−0.551325	−	0.834290i	$-0.685878\pi$
$29$	−5.93796			−1.10265			−0.551325	+	0.834290i	$0.685878\pi$
$30$		0			0
$31$	5.66037			1.01663			0.508316	−	0.861170i	$-0.330268\pi$
$31$	5.66037			1.01663			0.508316	+	0.861170i	$0.330268\pi$
$32$		0			0
$33$			3.68751i			0.641913i
$34$		0			0
$35$	−4.60263	−	8.49866i	−0.777986	−	1.43654i
$36$		0			0
$37$			11.2388i			1.84765i	0.382810	+	0.923827i	$0.374956\pi$
$37$			11.2388i			1.84765i	−0.382810	+	0.923827i	$0.625044\pi$
$38$		0			0
$39$	−4.58547			−0.734262
$40$		0			0
$41$	1.36096			0.212545			0.106273	−	0.994337i	$-0.466108\pi$
$41$	1.36096			0.212545			0.106273	+	0.994337i	$0.466108\pi$
$42$		0			0
$43$			6.90749i			1.05338i	0.850057	+	0.526691i	$0.176567\pi$
$43$			6.90749i			1.05338i	−0.850057	+	0.526691i	$0.823433\pi$
$44$		0			0
$45$	−1.96624	+	1.06486i	−0.293109	+	0.158739i
$46$		0			0
$47$			0.227862i			0.0332371i	0.999862	+	0.0166185i	$0.00529009\pi$
$47$			0.227862i			0.0332371i	−0.999862	+	0.0166185i	$0.994710\pi$
$48$		0			0
$49$	−11.6823			−1.66890
$50$		0			0
$51$	−7.89141			−1.10502
$52$		0			0
$53$		−	4.37470i		−	0.600911i	−0.953796	−	0.300456i	$-0.902861\pi$
$53$		−	4.37470i		−	0.600911i	0.953796	−	0.300456i	$-0.0971387\pi$
$54$		0			0
$55$	7.25051	−	3.92667i	0.977659	−	0.529472i
$56$		0			0
$57$			4.22571i			0.559709i
$58$		0			0
$59$	6.83840			0.890284			0.445142	−	0.895460i	$-0.353153\pi$
$59$	6.83840			0.890284			0.445142	+	0.895460i	$0.353153\pi$
$60$		0			0
$61$	4.41474			0.565249			0.282625	−	0.959231i	$-0.408795\pi$
$61$	4.41474			0.565249			0.282625	+	0.959231i	$0.408795\pi$
$62$		0			0
$63$			4.32230i			0.544559i
$64$		0			0
$65$	4.88286	+	9.01611i	0.605644	+	1.11831i
$66$		0			0
$67$			1.00000i			0.122169i
$67$			1.00000i			0.122169i
$68$		0			0
$69$	−1.04462			−0.125757
$70$		0			0
$71$	−6.97099			−0.827304			−0.413652	−	0.910435i	$-0.635747\pi$
$71$	−6.97099			−0.827304			−0.413652	+	0.910435i	$0.635747\pi$
$72$		0			0
$73$		−	1.49940i		−	0.175491i	−0.996143	−	0.0877455i	$-0.972034\pi$
$73$		−	1.49940i		−	0.175491i	0.996143	−	0.0877455i	$-0.0279662\pi$
$74$		0			0
$75$	4.18752	+	2.73216i	0.483533	+	0.315483i
$76$		0			0
$77$		−	15.9385i		−	1.81636i
$78$		0			0
$79$	−5.16488			−0.581094			−0.290547	−	0.956861i	$-0.593837\pi$
$79$	−5.16488			−0.581094			−0.290547	+	0.956861i	$0.593837\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			10.3758i			1.13889i	0.822030	+	0.569444i	$0.192842\pi$
$83$			10.3758i			1.13889i	−0.822030	+	0.569444i	$0.807158\pi$
$84$		0			0
$85$	8.40322	+	15.5164i	0.911457	+	1.68299i
$86$		0			0
$87$		−	5.93796i		−	0.636616i
$88$		0			0
$89$	2.71428			0.287713			0.143856	−	0.989599i	$-0.454050\pi$
$89$	2.71428			0.287713			0.143856	+	0.989599i	$0.454050\pi$
$90$		0			0
$91$	19.8198			2.07768
$92$		0			0
$93$			5.66037i			0.586953i
$94$		0			0
$95$	8.30873	−	4.49977i	0.852458	−	0.461667i
$96$		0			0
$97$		−	8.15818i		−	0.828338i	−0.910200	−	0.414169i	$-0.864072\pi$
$97$		−	8.15818i		−	0.828338i	0.910200	−	0.414169i	$-0.135928\pi$
$98$		0			0
$99$	−3.68751			−0.370609
$100$		0			0
$101$	2.75538			0.274171			0.137085	−	0.990559i	$-0.456227\pi$
$101$	2.75538			0.274171			0.137085	+	0.990559i	$0.456227\pi$
$102$		0			0
$103$			11.5307i			1.13615i	0.822975	+	0.568077i	$0.192312\pi$
$103$			11.5307i			1.13615i	−0.822975	+	0.568077i	$0.807688\pi$
$104$		0			0
$105$	8.49866	−	4.60263i	0.829384	−	0.449170i
$106$		0			0
$107$		−	12.8485i		−	1.24211i	−0.783767	−	0.621055i	$-0.786704\pi$
$107$		−	12.8485i		−	1.24211i	0.783767	−	0.621055i	$-0.213296\pi$
$108$		0			0
$109$	3.68327			0.352793			0.176397	−	0.984319i	$-0.443556\pi$
$109$	3.68327			0.352793			0.176397	+	0.984319i	$0.443556\pi$
$110$		0			0
$111$	−11.2388			−1.06674
$112$		0			0
$113$		−	7.93123i		−	0.746108i	−0.927810	−	0.373054i	$-0.878311\pi$
$113$		−	7.93123i		−	0.746108i	0.927810	−	0.373054i	$-0.121689\pi$
$114$		0			0
$115$	1.11237	+	2.05396i	0.103729	+	0.191533i
$116$		0			0
$117$		−	4.58547i		−	0.423926i
$118$		0			0
$119$	34.1091			3.12677
$120$		0			0
$121$	2.59773			0.236157
$122$		0			0
$123$			1.36096i			0.122713i
$124$		0			0
$125$	0.912975	−	11.1430i	0.0816590	−	0.996660i
$126$		0			0
$127$		−	6.32450i		−	0.561208i	−0.959824	−	0.280604i	$-0.909465\pi$
$127$		−	6.32450i		−	0.561208i	0.959824	−	0.280604i	$-0.0905348\pi$
$128$		0			0
$129$	−6.90749			−0.608170
$130$		0			0
$131$	8.81351			0.770040			0.385020	−	0.922908i	$-0.374195\pi$
$131$	8.81351			0.770040			0.385020	+	0.922908i	$0.374195\pi$
$132$		0			0
$133$		−	18.2648i		−	1.58376i
$134$		0			0
$135$	−1.06486	−	1.96624i	−0.0916482	−	0.169227i
$136$		0			0
$137$		−	8.13167i		−	0.694735i	−0.937729	−	0.347368i	$-0.887076\pi$
$137$		−	8.13167i		−	0.694735i	0.937729	−	0.347368i	$-0.112924\pi$
$138$		0			0
$139$	4.35176			0.369111			0.184556	−	0.982822i	$-0.440915\pi$
$139$	4.35176			0.369111			0.184556	+	0.982822i	$0.440915\pi$
$140$		0			0
$141$	−0.227862			−0.0191894
$142$		0			0
$143$			16.9089i			1.41400i
$144$		0			0
$145$	−11.6754	+	6.32307i	−0.969591	+	0.525102i
$146$		0			0
$147$		−	11.6823i		−	0.963538i
$148$		0			0
$149$	5.32537			0.436271			0.218136	−	0.975918i	$-0.430002\pi$
$149$	5.32537			0.436271			0.218136	+	0.975918i	$0.430002\pi$
$150$		0			0
$151$	14.9734			1.21852			0.609260	−	0.792970i	$-0.291466\pi$
$151$	14.9734			1.21852			0.609260	+	0.792970i	$0.291466\pi$
$152$		0			0
$153$		−	7.89141i		−	0.637983i
$154$		0			0
$155$	11.1296	−	6.02748i	0.893953	−	0.484139i
$156$		0			0
$157$		−	23.0196i		−	1.83717i	−0.395227	−	0.918583i	$-0.629334\pi$
$157$		−	23.0196i		−	1.83717i	0.395227	−	0.918583i	$-0.370666\pi$
$158$		0			0
$159$	4.37470			0.346936
$160$		0			0
$161$	4.51515			0.355844
$162$		0			0
$163$		−	19.1669i		−	1.50127i	−0.660719	−	0.750633i	$-0.729749\pi$
$163$		−	19.1669i		−	1.50127i	0.660719	−	0.750633i	$-0.270251\pi$
$164$		0			0
$165$	3.92667	+	7.25051i	0.305691	+	0.564452i
$166$		0			0
$167$		−	14.7151i		−	1.13869i	−0.822099	−	0.569345i	$-0.807197\pi$
$167$		−	14.7151i		−	1.13869i	0.822099	−	0.569345i	$-0.192803\pi$
$168$		0			0
$169$	−8.02650			−0.617423
$170$		0			0
$171$	−4.22571			−0.323148
$172$		0			0
$173$			14.7906i			1.12451i	0.826965	+	0.562253i	$0.190065\pi$
$173$			14.7906i			1.12451i	−0.826965	+	0.562253i	$0.809935\pi$
$174$		0			0
$175$	−18.0997	−	11.8092i	−1.36821	−	0.892694i
$176$		0			0
$177$			6.83840i			0.514006i
$178$		0			0
$179$	−16.7747			−1.25380			−0.626898	−	0.779101i	$-0.715676\pi$
$179$	−16.7747			−1.25380			−0.626898	+	0.779101i	$0.715676\pi$
$180$		0			0
$181$	18.8727			1.40279			0.701397	−	0.712771i	$-0.252560\pi$
$181$	18.8727			1.40279			0.701397	+	0.712771i	$0.252560\pi$
$182$		0			0
$183$			4.41474i			0.326347i
$184$		0			0
$185$	11.9677	+	22.0982i	0.879886	+	1.62469i
$186$		0			0
$187$			29.0997i			2.12798i
$188$		0			0
$189$	−4.32230			−0.314401
$190$		0			0
$191$	4.24583			0.307217			0.153609	−	0.988132i	$-0.450910\pi$
$191$	4.24583			0.307217			0.153609	+	0.988132i	$0.450910\pi$
$192$		0			0
$193$		−	17.8178i		−	1.28255i	−0.767311	−	0.641275i	$-0.778406\pi$
$193$		−	17.8178i		−	1.28255i	0.767311	−	0.641275i	$-0.221594\pi$
$194$		0			0
$195$	−9.01611	+	4.88286i	−0.645657	+	0.349669i
$196$		0			0
$197$			7.79528i			0.555391i	0.960669	+	0.277696i	$0.0895706\pi$
$197$			7.79528i			0.555391i	−0.960669	+	0.277696i	$0.910429\pi$
$198$		0			0
$199$	1.01866			0.0722111			0.0361055	−	0.999348i	$-0.488505\pi$
$199$	1.01866			0.0722111			0.0361055	+	0.999348i	$0.488505\pi$
$200$		0			0
$201$	−1.00000			−0.0705346
$202$		0			0
$203$			25.6656i			1.80137i
$204$		0			0
$205$	2.67596	−	1.44922i	0.186897	−	0.101218i
$206$		0			0
$207$		−	1.04462i		−	0.0726059i
$208$		0			0
$209$	15.5823			1.07785
$210$		0			0
$211$	16.9281			1.16538			0.582689	−	0.812695i	$-0.302001\pi$
$211$	16.9281			1.16538			0.582689	+	0.812695i	$0.302001\pi$
$212$		0			0
$213$		−	6.97099i		−	0.477644i
$214$		0			0
$215$	7.35548	+	13.5817i	0.501640	+	0.926267i
$216$		0			0
$217$		−	24.4658i		−	1.66085i
$218$		0			0
$219$	1.49940			0.101320
$220$		0			0
$221$	−36.1858			−2.43412
$222$		0			0
$223$			19.6720i			1.31733i	0.752435	+	0.658666i	$0.228879\pi$
$223$			19.6720i			1.31733i	−0.752435	+	0.658666i	$0.771121\pi$
$224$		0			0
$225$	−2.73216	+	4.18752i	−0.182144	+	0.279168i
$226$		0			0
$227$			21.8906i			1.45293i	0.687203	+	0.726466i	$0.258838\pi$
$227$			21.8906i			1.45293i	−0.687203	+	0.726466i	$0.741162\pi$
$228$		0			0
$229$	−3.28727			−0.217229			−0.108614	−	0.994084i	$-0.534641\pi$
$229$	−3.28727			−0.217229			−0.108614	+	0.994084i	$0.534641\pi$
$230$		0			0
$231$	15.9385			1.04868
$232$		0			0
$233$			10.4772i			0.686382i	0.939266	+	0.343191i	$0.111508\pi$
$233$			10.4772i			0.686382i	−0.939266	+	0.343191i	$0.888492\pi$
$234$		0			0
$235$	0.242640	+	0.448030i	0.0158281	+	0.0292263i
$236$		0			0
$237$		−	5.16488i		−	0.335495i
$238$		0			0
$239$	−19.8748			−1.28559			−0.642796	−	0.766038i	$-0.722226\pi$
$239$	−19.8748			−1.28559			−0.642796	+	0.766038i	$0.722226\pi$
$240$		0			0
$241$	−11.2871			−0.727065			−0.363533	−	0.931581i	$-0.618429\pi$
$241$	−11.2871			−0.727065			−0.363533	+	0.931581i	$0.618429\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−22.9701	+	12.4399i	−1.46751	+	0.794759i
$246$		0			0
$247$			19.3768i			1.23292i
$248$		0			0
$249$	−10.3758			−0.657537
$250$		0			0
$251$	−30.1154			−1.90087			−0.950434	−	0.310928i	$-0.899360\pi$
$251$	−30.1154			−1.90087			−0.950434	+	0.310928i	$0.899360\pi$
$252$		0			0
$253$			3.85204i			0.242175i
$254$		0			0
$255$	−15.5164	+	8.40322i	−0.971673	+	0.526230i
$256$		0			0
$257$			26.5811i			1.65808i	0.559186	+	0.829042i	$0.311114\pi$
$257$			26.5811i			1.65808i	−0.559186	+	0.829042i	$0.688886\pi$
$258$		0			0
$259$	48.5776			3.01847
$260$		0			0
$261$	5.93796			0.367550
$262$		0			0
$263$			10.5190i			0.648629i	0.945949	+	0.324315i	$0.105134\pi$
$263$			10.5190i			0.648629i	−0.945949	+	0.324315i	$0.894866\pi$
$264$		0			0
$265$	−4.65843	−	8.60169i	−0.286165	−	0.528398i
$266$		0			0
$267$			2.71428i			0.166111i
$268$		0			0
$269$	0.906058			0.0552433			0.0276217	−	0.999618i	$-0.491207\pi$
$269$	0.906058			0.0552433			0.0276217	+	0.999618i	$0.491207\pi$
$270$		0			0
$271$	11.0749			0.672750			0.336375	−	0.941728i	$-0.390799\pi$
$271$	11.0749			0.672750			0.336375	+	0.941728i	$0.390799\pi$
$272$		0			0
$273$			19.8198i			1.19955i
$274$		0			0
$275$	10.0749	−	15.4415i	0.607538	−	0.931158i
$276$		0			0
$277$			3.74825i			0.225211i	0.993640	+	0.112605i	$0.0359196\pi$
$277$			3.74825i			0.225211i	−0.993640	+	0.112605i	$0.964080\pi$
$278$		0			0
$279$	−5.66037			−0.338878
$280$		0			0
$281$	−17.7546			−1.05915			−0.529576	−	0.848263i	$-0.677649\pi$
$281$	−17.7546			−1.05915			−0.529576	+	0.848263i	$0.677649\pi$
$282$		0			0
$283$			19.6755i			1.16959i	0.811183	+	0.584793i	$0.198824\pi$
$283$			19.6755i			1.16959i	−0.811183	+	0.584793i	$0.801176\pi$
$284$		0			0
$285$	4.49977	+	8.30873i	0.266543	+	0.492167i
$286$		0			0
$287$		−	5.88246i		−	0.347230i
$288$		0			0
$289$	−45.2744			−2.66320
$290$		0			0
$291$	8.15818			0.478241
$292$		0			0
$293$		−	16.3514i		−	0.955259i	−0.878561	−	0.477629i	$-0.841496\pi$
$293$		−	16.3514i		−	0.955259i	0.878561	−	0.477629i	$-0.158504\pi$
$294$		0			0
$295$	13.4459	−	7.28191i	0.782851	−	0.423969i
$296$		0			0
$297$		−	3.68751i		−	0.213971i
$298$		0			0
$299$	−4.79006			−0.277016
$300$		0			0
$301$	29.8562			1.72088
$302$		0			0
$303$			2.75538i			0.158293i
$304$		0			0
$305$	8.68042	−	4.70106i	0.497039	−	0.269182i
$306$		0			0
$307$		−	0.589802i		−	0.0336618i	−0.999858	−	0.0168309i	$-0.994642\pi$
$307$		−	0.589802i		−	0.0336618i	0.999858	−	0.0168309i	$-0.00535770\pi$
$308$		0			0
$309$	−11.5307			−0.655959
$310$		0			0
$311$	13.3850			0.758994			0.379497	−	0.925193i	$-0.376097\pi$
$311$	13.3850			0.758994			0.379497	+	0.925193i	$0.376097\pi$
$312$		0			0
$313$		−	15.9356i		−	0.900735i	−0.892843	−	0.450368i	$-0.851293\pi$
$313$		−	15.9356i		−	0.900735i	0.892843	−	0.450368i	$-0.148707\pi$
$314$		0			0
$315$	4.60263	+	8.49866i	0.259329	+	0.478845i
$316$		0			0
$317$		−	16.4627i		−	0.924640i	−0.886713	−	0.462320i	$-0.847017\pi$
$317$		−	16.4627i		−	0.924640i	0.886713	−	0.462320i	$-0.152983\pi$
$318$		0			0
$319$	−21.8963			−1.22596
$320$		0			0
$321$	12.8485			0.717133
$322$		0			0
$323$			33.3468i			1.85547i
$324$		0			0
$325$	19.2017	+	12.5282i	1.06512	+	0.694942i
$326$		0			0
$327$			3.68327i			0.203685i
$328$		0			0
$329$	0.984887			0.0542986
$330$		0			0
$331$	32.0995			1.76435			0.882175	−	0.470923i	$-0.156079\pi$
$331$	32.0995			1.76435			0.882175	+	0.470923i	$0.156079\pi$
$332$		0			0
$333$		−	11.2388i		−	0.615885i
$334$		0			0
$335$	1.06486	+	1.96624i	0.0581793	+	0.107427i
$336$		0			0
$337$		−	2.03842i		−	0.111040i	−0.998458	−	0.0555198i	$-0.982318\pi$
$337$		−	2.03842i		−	0.111040i	0.998458	−	0.0555198i	$-0.0176816\pi$
$338$		0			0
$339$	7.93123			0.430766
$340$		0			0
$341$	20.8727			1.13032
$342$		0			0
$343$			20.2382i			1.09276i
$344$		0			0
$345$	−2.05396	+	1.11237i	−0.110582	+	0.0598878i
$346$		0			0
$347$			16.6466i			0.893639i	0.894624	+	0.446819i	$0.147443\pi$
$347$			16.6466i			0.893639i	−0.894624	+	0.446819i	$0.852557\pi$
$348$		0			0
$349$	21.7609			1.16484			0.582418	−	0.812889i	$-0.302107\pi$
$349$	21.7609			1.16484			0.582418	+	0.812889i	$0.302107\pi$
$350$		0			0
$351$	4.58547			0.244754
$352$		0			0
$353$			12.5331i			0.667069i	0.942738	+	0.333535i	$0.108241\pi$
$353$			12.5331i			0.667069i	−0.942738	+	0.333535i	$0.891759\pi$
$354$		0			0
$355$	−13.7066	+	7.42310i	−0.727471	+	0.393977i
$356$		0			0
$357$			34.1091i			1.80524i
$358$		0			0
$359$	−21.5144			−1.13549			−0.567743	−	0.823206i	$-0.692183\pi$
$359$	−21.5144			−1.13549			−0.567743	+	0.823206i	$0.692183\pi$
$360$		0			0
$361$	−1.14340			−0.0601789
$362$		0			0
$363$			2.59773i			0.136345i
$364$		0			0
$365$	−1.59664	−	2.94816i	−0.0835720	−	0.154314i
$366$		0			0
$367$		−	13.6410i		−	0.712052i	−0.934476	−	0.356026i	$-0.884131\pi$
$367$		−	13.6410i		−	0.712052i	0.934476	−	0.356026i	$-0.115869\pi$
$368$		0			0
$369$	−1.36096			−0.0708485
$370$		0			0
$371$	−18.9088			−0.981694
$372$		0			0
$373$		−	29.8002i		−	1.54300i	−0.636231	−	0.771498i	$-0.719508\pi$
$373$		−	29.8002i		−	1.54300i	0.636231	−	0.771498i	$-0.280492\pi$
$374$		0			0
$375$	11.1430	+	0.912975i	0.575422	+	0.0471458i
$376$		0			0
$377$		−	27.2283i		−	1.40233i
$378$		0			0
$379$	−30.1563			−1.54902			−0.774512	−	0.632559i	$-0.782005\pi$
$379$	−30.1563			−1.54902			−0.774512	+	0.632559i	$0.782005\pi$
$380$		0			0
$381$	6.32450			0.324014
$382$		0			0
$383$		−	5.16543i		−	0.263941i	−0.991254	−	0.131971i	$-0.957870\pi$
$383$		−	5.16543i		−	0.263941i	0.991254	−	0.131971i	$-0.0421305\pi$
$384$		0			0
$385$	−16.9722	−	31.3389i	−0.864985	−	1.59718i
$386$		0			0
$387$		−	6.90749i		−	0.351127i
$388$		0			0
$389$	24.9691			1.26598			0.632992	−	0.774159i	$-0.281827\pi$
$389$	24.9691			1.26598			0.632992	+	0.774159i	$0.281827\pi$
$390$		0			0
$391$	−8.24351			−0.416892
$392$		0			0
$393$			8.81351i			0.444583i
$394$		0			0
$395$	−10.1554	+	5.49985i	−0.510972	+	0.276727i
$396$		0			0
$397$		−	33.8747i		−	1.70012i	−0.526685	−	0.850060i	$-0.676565\pi$
$397$		−	33.8747i		−	1.70012i	0.526685	−	0.850060i	$-0.323435\pi$
$398$		0			0
$399$	18.2648			0.914382
$400$		0			0
$401$	−37.5275			−1.87403			−0.937016	−	0.349286i	$-0.886424\pi$
$401$	−37.5275			−1.87403			−0.937016	+	0.349286i	$0.886424\pi$
$402$		0			0
$403$			25.9554i			1.29293i
$404$		0			0
$405$	1.96624	−	1.06486i	0.0977030	−	0.0529131i
$406$		0			0
$407$			41.4433i			2.05427i
$408$		0			0
$409$	−38.5255			−1.90496			−0.952481	−	0.304597i	$-0.901478\pi$
$409$	−38.5255			−1.90496			−0.952481	+	0.304597i	$0.901478\pi$
$410$		0			0
$411$	8.13167			0.401106
$412$		0			0
$413$		−	29.5576i		−	1.45444i
$414$		0			0
$415$	11.0487	+	20.4012i	0.542359	+	1.00146i
$416$		0			0
$417$			4.35176i			0.213107i
$418$		0			0
$419$	−3.95992			−0.193455			−0.0967274	−	0.995311i	$-0.530837\pi$
$419$	−3.95992			−0.193455			−0.0967274	+	0.995311i	$0.530837\pi$
$420$		0			0
$421$	−24.5101			−1.19455			−0.597274	−	0.802038i	$-0.703749\pi$
$421$	−24.5101			−1.19455			−0.597274	+	0.802038i	$0.703749\pi$
$422$		0			0
$423$		−	0.227862i		−	0.0110790i
$424$		0			0
$425$	33.0454	+	21.5606i	1.60294	+	1.04584i
$426$		0			0
$427$		−	19.0818i		−	0.923434i
$428$		0			0
$429$	−16.9089			−0.816372
$430$		0			0
$431$	−37.0414			−1.78422			−0.892110	−	0.451819i	$-0.850775\pi$
$431$	−37.0414			−1.78422			−0.892110	+	0.451819i	$0.850775\pi$
$432$		0			0
$433$		−	12.3638i		−	0.594164i	−0.954852	−	0.297082i	$-0.903986\pi$
$433$		−	12.3638i		−	0.594164i	0.954852	−	0.297082i	$-0.0960136\pi$
$434$		0			0
$435$	−6.32307	−	11.6754i	−0.303168	−	0.559794i
$436$		0			0
$437$			4.41425i			0.211162i
$438$		0			0
$439$	15.0541			0.718492			0.359246	−	0.933243i	$-0.383034\pi$
$439$	15.0541			0.718492			0.359246	+	0.933243i	$0.383034\pi$
$440$		0			0
$441$	11.6823			0.556299
$442$		0			0
$443$		−	24.4444i		−	1.16139i	−0.814122	−	0.580694i	$-0.802781\pi$
$443$		−	24.4444i		−	1.16139i	0.814122	−	0.580694i	$-0.197219\pi$
$444$		0			0
$445$	5.33691	−	2.89032i	0.252994	−	0.137014i
$446$		0			0
$447$			5.32537i			0.251881i
$448$		0			0
$449$	−1.53808			−0.0725864			−0.0362932	−	0.999341i	$-0.511555\pi$
$449$	−1.53808			−0.0725864			−0.0362932	+	0.999341i	$0.511555\pi$
$450$		0			0
$451$	5.01854			0.236314
$452$		0			0
$453$			14.9734i			0.703513i
$454$		0			0
$455$	38.9703	−	21.1052i	1.82696	−	0.989427i
$456$		0			0
$457$			17.2881i			0.808705i	0.914603	+	0.404353i	$0.132503\pi$
$457$			17.2881i			0.808705i	−0.914603	+	0.404353i	$0.867497\pi$
$458$		0			0
$459$	7.89141			0.368340
$460$		0			0
$461$	39.7237			1.85012			0.925058	−	0.379826i	$-0.124016\pi$
$461$	39.7237			1.85012			0.925058	+	0.379826i	$0.124016\pi$
$462$		0			0
$463$		−	21.7437i		−	1.01051i	−0.862969	−	0.505257i	$-0.831398\pi$
$463$		−	21.7437i		−	1.01051i	0.862969	−	0.505257i	$-0.168602\pi$
$464$		0			0
$465$	6.02748	+	11.1296i	0.279518	+	0.516124i
$466$		0			0
$467$			17.1176i			0.792108i	0.918227	+	0.396054i	$0.129621\pi$
$467$			17.1176i			0.792108i	−0.918227	+	0.396054i	$0.870379\pi$
$468$		0			0
$469$	4.32230			0.199585
$470$		0			0
$471$	23.0196			1.06069
$472$		0			0
$473$			25.4714i			1.17118i
$474$		0			0
$475$	11.5453	−	17.6952i	0.529736	−	0.811912i
$476$		0			0
$477$			4.37470i			0.200304i
$478$		0			0
$479$	33.0239			1.50890			0.754449	−	0.656358i	$-0.227904\pi$
$479$	33.0239			1.50890			0.754449	+	0.656358i	$0.227904\pi$
$480$		0			0
$481$	−51.5353			−2.34981
$482$		0			0
$483$			4.51515i			0.205446i
$484$		0			0
$485$	−8.68729	−	16.0409i	−0.394470	−	0.728380i
$486$		0			0
$487$		−	26.9318i		−	1.22040i	−0.792249	−	0.610198i	$-0.791090\pi$
$487$		−	26.9318i		−	1.22040i	0.792249	−	0.610198i	$-0.208910\pi$
$488$		0			0
$489$	19.1669			0.866756
$490$		0			0
$491$	26.7624			1.20777			0.603886	−	0.797071i	$-0.293618\pi$
$491$	26.7624			1.20777			0.603886	+	0.797071i	$0.293618\pi$
$492$		0			0
$493$		−	46.8589i		−	2.11042i
$494$		0			0
$495$	−7.25051	+	3.92667i	−0.325886	+	0.176491i
$496$		0			0
$497$			30.1307i			1.35155i
$498$		0			0
$499$	18.7069			0.837435			0.418717	−	0.908117i	$-0.362480\pi$
$499$	18.7069			0.837435			0.418717	+	0.908117i	$0.362480\pi$
$500$		0			0
$501$	14.7151			0.657423
$502$		0			0
$503$		−	37.1869i		−	1.65808i	−0.559189	−	0.829040i	$-0.688887\pi$
$503$		−	37.1869i		−	1.65808i	0.559189	−	0.829040i	$-0.311113\pi$
$504$		0			0
$505$	5.41773	−	2.93409i	0.241086	−	0.130565i
$506$		0			0
$507$		−	8.02650i		−	0.356469i
$508$		0			0
$509$	−5.39023			−0.238918			−0.119459	−	0.992839i	$-0.538116\pi$
$509$	−5.39023			−0.238918			−0.119459	+	0.992839i	$0.538116\pi$
$510$		0			0
$511$	−6.48084			−0.286695
$512$		0			0
$513$		−	4.22571i		−	0.186570i
$514$		0			0
$515$	12.2785	+	22.6721i	0.541057	+	0.999051i
$516$		0			0
$517$			0.840243i			0.0369538i
$518$		0			0
$519$	−14.7906			−0.649234
$520$		0			0
$521$	−18.7511			−0.821500			−0.410750	−	0.911748i	$-0.634733\pi$
$521$	−18.7511			−0.821500			−0.410750	+	0.911748i	$0.634733\pi$
$522$		0			0
$523$		−	1.43214i		−	0.0626230i	−0.999510	−	0.0313115i	$-0.990032\pi$
$523$		−	1.43214i		−	0.0626230i	0.999510	−	0.0313115i	$-0.00996839\pi$
$524$		0			0
$525$	11.8092	−	18.0997i	0.515397	−	0.789936i
$526$		0			0
$527$			44.6683i			1.94578i
$528$		0			0
$529$	21.9088			0.952555
$530$		0			0
$531$	−6.83840			−0.296761
$532$		0			0
$533$			6.24061i			0.270311i
$534$		0			0
$535$	−13.6818	−	25.2631i	−0.591515	−	1.09222i
$536$		0			0
$537$		−	16.7747i		−	0.723880i
$538$		0			0
$539$	−43.0785			−1.85552
$540$		0			0
$541$	2.35121			0.101087			0.0505433	−	0.998722i	$-0.483905\pi$
$541$	2.35121			0.101087			0.0505433	+	0.998722i	$0.483905\pi$
$542$		0			0
$543$			18.8727i			0.809904i
$544$		0			0
$545$	7.24218	−	3.92215i	0.310221	−	0.168007i
$546$		0			0
$547$			41.0405i			1.75476i	0.479793	+	0.877382i	$0.340712\pi$
$547$			41.0405i			1.75476i	−0.479793	+	0.877382i	$0.659288\pi$
$548$		0			0
$549$	−4.41474			−0.188416
$550$		0			0
$551$	−25.0921			−1.06896
$552$		0			0
$553$			22.3241i			0.949319i
$554$		0			0
$555$	−22.0982	+	11.9677i	−0.938017	+	0.508003i
$556$		0			0
$557$			3.09843i			0.131285i	0.997843	+	0.0656424i	$0.0209097\pi$
$557$			3.09843i			0.131285i	−0.997843	+	0.0656424i	$0.979090\pi$
$558$		0			0
$559$	−31.6740			−1.33967
$560$		0			0
$561$	−29.0997			−1.22859
$562$		0			0
$563$		−	15.9187i		−	0.670891i	−0.942060	−	0.335446i	$-0.891113\pi$
$563$		−	15.9187i		−	0.670891i	0.942060	−	0.335446i	$-0.108887\pi$
$564$		0			0
$565$	−8.44562	−	15.5947i	−0.355310	−	0.656073i
$566$		0			0
$567$		−	4.32230i		−	0.181520i
$568$		0			0
$569$	28.1166			1.17871			0.589355	−	0.807874i	$-0.299382\pi$
$569$	28.1166			1.17871			0.589355	+	0.807874i	$0.299382\pi$
$570$		0			0
$571$	−21.2535			−0.889431			−0.444716	−	0.895672i	$-0.646695\pi$
$571$	−21.2535			−0.889431			−0.444716	+	0.895672i	$0.646695\pi$
$572$		0			0
$573$			4.24583i			0.177372i
$574$		0			0
$575$	4.37435	+	2.85406i	0.182423	+	0.119023i
$576$		0			0
$577$			44.9336i			1.87061i	0.353842	+	0.935305i	$0.384875\pi$
$577$			44.9336i			1.87061i	−0.353842	+	0.935305i	$0.615125\pi$
$578$		0			0
$579$	17.8178			0.740480
$580$		0			0
$581$	44.8472			1.86057
$582$		0			0
$583$		−	16.1317i		−	0.668109i
$584$		0			0
$585$	−4.88286	−	9.01611i	−0.201881	−	0.372770i
$586$		0			0
$587$			14.3953i			0.594157i	0.954853	+	0.297078i	$0.0960123\pi$
$587$			14.3953i			0.594157i	−0.954853	+	0.297078i	$0.903988\pi$
$588$		0			0
$589$	23.9191			0.985568
$590$		0			0
$591$	−7.79528			−0.320655
$592$		0			0
$593$		−	15.5990i		−	0.640576i	−0.947320	−	0.320288i	$-0.896220\pi$
$593$		−	15.5990i		−	0.640576i	0.947320	−	0.320288i	$-0.103780\pi$
$594$		0			0
$595$	67.0664	−	36.3212i	2.74946	−	1.48903i
$596$		0			0
$597$			1.01866i			0.0416911i
$598$		0			0
$599$	−8.80305			−0.359683			−0.179841	−	0.983696i	$-0.557558\pi$
$599$	−8.80305			−0.359683			−0.179841	+	0.983696i	$0.557558\pi$
$600$		0			0
$601$	−5.90124			−0.240717			−0.120358	−	0.992731i	$-0.538404\pi$
$601$	−5.90124			−0.240717			−0.120358	+	0.992731i	$0.538404\pi$
$602$		0			0
$603$		−	1.00000i		−	0.0407231i
$604$		0			0
$605$	5.10774	−	2.76621i	0.207659	−	0.112462i
$606$		0			0
$607$		−	42.2861i		−	1.71634i	−0.513367	−	0.858169i	$-0.671602\pi$
$607$		−	42.2861i		−	1.71634i	0.513367	−	0.858169i	$-0.328398\pi$
$608$		0			0
$609$	−25.6656			−1.04002
$610$		0			0
$611$	−1.04485			−0.0422702
$612$		0			0
$613$		−	18.3331i		−	0.740469i	−0.928938	−	0.370234i	$-0.879277\pi$
$613$		−	18.3331i		−	0.740469i	0.928938	−	0.370234i	$-0.120723\pi$
$614$		0			0
$615$	1.44922	+	2.67596i	0.0584382	+	0.107905i
$616$		0			0
$617$			21.0364i			0.846892i	0.905921	+	0.423446i	$0.139180\pi$
$617$			21.0364i			0.846892i	−0.905921	+	0.423446i	$0.860820\pi$
$618$		0			0
$619$	−8.00519			−0.321756			−0.160878	−	0.986974i	$-0.551433\pi$
$619$	−8.00519			−0.321756			−0.160878	+	0.986974i	$0.551433\pi$
$620$		0			0
$621$	1.04462			0.0419190
$622$		0			0
$623$		−	11.7319i		−	0.470030i
$624$		0			0
$625$	−10.0706	−	22.8820i	−0.402823	−	0.915278i
$626$		0			0
$627$			15.5823i			0.622299i
$628$		0			0
$629$	−88.6903			−3.53632
$630$		0			0
$631$	38.5012			1.53271			0.766354	−	0.642418i	$-0.222069\pi$
$631$	38.5012			1.53271			0.766354	+	0.642418i	$0.222069\pi$
$632$		0			0
$633$			16.9281i			0.672831i
$634$		0			0
$635$	−6.73468	−	12.4355i	−0.267258	−	0.493486i
$636$		0			0
$637$		−	53.5687i		−	2.12247i
$638$		0			0
$639$	6.97099			0.275768
$640$		0			0
$641$	−36.6360			−1.44703			−0.723517	−	0.690307i	$-0.757476\pi$
$641$	−36.6360			−1.44703			−0.723517	+	0.690307i	$0.757476\pi$
$642$		0			0
$643$		−	0.482014i		−	0.0190088i	−0.999955	−	0.00950439i	$-0.996975\pi$
$643$		−	0.482014i		−	0.0190088i	0.999955	−	0.00950439i	$-0.00302539\pi$
$644$		0			0
$645$	−13.5817	+	7.35548i	−0.534781	+	0.289622i
$646$		0			0
$647$		−	42.4176i		−	1.66761i	−0.552060	−	0.833804i	$-0.686158\pi$
$647$		−	42.4176i		−	1.66761i	0.552060	−	0.833804i	$-0.313842\pi$
$648$		0			0
$649$	25.2167			0.989841
$650$		0			0
$651$	24.4658			0.958891
$652$		0			0
$653$		−	1.57080i		−	0.0614703i	−0.999528	−	0.0307352i	$-0.990215\pi$
$653$		−	1.57080i		−	0.0614703i	0.999528	−	0.0307352i	$-0.00978485\pi$
$654$		0			0
$655$	17.3294	−	9.38512i	0.677117	−	0.366707i
$656$		0			0
$657$			1.49940i			0.0584970i
$658$		0			0
$659$	1.85440			0.0722370			0.0361185	−	0.999348i	$-0.488501\pi$
$659$	1.85440			0.0722370			0.0361185	+	0.999348i	$0.488501\pi$
$660$		0			0
$661$	17.8088			0.692683			0.346342	−	0.938108i	$-0.387424\pi$
$661$	17.8088			0.692683			0.346342	+	0.938108i	$0.387424\pi$
$662$		0			0
$663$		−	36.1858i		−	1.40534i
$664$		0			0
$665$	−19.4494	−	35.9128i	−0.754214	−	1.39264i
$666$		0			0
$667$		−	6.20289i		−	0.240177i
$668$		0			0
$669$	−19.6720			−0.760562
$670$		0			0
$671$	16.2794			0.628459
$672$		0			0
$673$			8.76791i			0.337978i	0.985618	+	0.168989i	$0.0540502\pi$
$673$			8.76791i			0.337978i	−0.985618	+	0.168989i	$0.945950\pi$
$674$		0			0
$675$	−4.18752	−	2.73216i	−0.161178	−	0.105161i
$676$		0			0
$677$			14.2467i			0.547547i	0.961794	+	0.273773i	$0.0882718\pi$
$677$			14.2467i			0.547547i	−0.961794	+	0.273773i	$0.911728\pi$
$678$		0			0
$679$	−35.2621			−1.35324
$680$		0			0
$681$	−21.8906			−0.838850
$682$		0			0
$683$		−	7.19495i		−	0.275307i	−0.990480	−	0.137654i	$-0.956044\pi$
$683$		−	7.19495i		−	0.275307i	0.990480	−	0.137654i	$-0.0439561\pi$
$684$		0			0
$685$	−8.65905	−	15.9888i	−0.330845	−	0.610899i
$686$		0			0
$687$		−	3.28727i		−	0.125417i
$688$		0			0
$689$	20.0600			0.764227
$690$		0			0
$691$	10.2758			0.390911			0.195455	−	0.980713i	$-0.437382\pi$
$691$	10.2758			0.390911			0.195455	+	0.980713i	$0.437382\pi$
$692$		0			0
$693$			15.9385i			0.605454i
$694$		0			0
$695$	8.55658	−	4.63400i	0.324570	−	0.175778i
$696$		0			0
$697$			10.7399i			0.406801i
$698$		0			0
$699$	−10.4772			−0.396283
$700$		0			0
$701$	4.73210			0.178729			0.0893644	−	0.995999i	$-0.471516\pi$
$701$	4.73210			0.178729			0.0893644	+	0.995999i	$0.471516\pi$
$702$		0			0
$703$			47.4920i			1.79120i
$704$		0			0
$705$	−0.448030	+	0.242640i	−0.0168738	+	0.00913835i
$706$		0			0
$707$		−	11.9096i		−	0.447906i
$708$		0			0
$709$	−23.8274			−0.894858			−0.447429	−	0.894319i	$-0.647660\pi$
$709$	−23.8274			−0.894858			−0.447429	+	0.894319i	$0.647660\pi$
$710$		0			0
$711$	5.16488			0.193698
$712$		0			0
$713$			5.91292i			0.221441i
$714$		0			0
$715$	18.0056	+	33.2470i	0.673371	+	1.24337i
$716$		0			0
$717$		−	19.8748i		−	0.742236i
$718$		0			0
$719$	5.11082			0.190601			0.0953007	−	0.995449i	$-0.469619\pi$
$719$	5.11082			0.190601			0.0953007	+	0.995449i	$0.469619\pi$
$720$		0			0
$721$	49.8391			1.85611
$722$		0			0
$723$		−	11.2871i		−	0.419771i
$724$		0			0
$725$	−16.2235	+	24.8653i	−0.602524	+	0.923474i
$726$		0			0
$727$		−	30.2795i		−	1.12300i	−0.827475	−	0.561502i	$-0.810224\pi$
$727$		−	30.2795i		−	1.12300i	0.827475	−	0.561502i	$-0.189776\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−54.5098			−2.01612
$732$		0			0
$733$		−	21.5837i		−	0.797213i	−0.917122	−	0.398607i	$-0.869494\pi$
$733$		−	21.5837i		−	0.797213i	0.917122	−	0.398607i	$-0.130506\pi$
$734$		0			0
$735$	−12.4399	−	22.9701i	−0.458854	−	0.847265i
$736$		0			0
$737$			3.68751i			0.135831i
$738$		0			0
$739$	−51.2289			−1.88449			−0.942243	−	0.334929i	$-0.891288\pi$
$739$	−51.2289			−1.88449			−0.942243	+	0.334929i	$0.891288\pi$
$740$		0			0
$741$	−19.3768			−0.711826
$742$		0			0
$743$		−	42.0409i		−	1.54233i	−0.636634	−	0.771166i	$-0.719674\pi$
$743$		−	42.0409i		−	1.54233i	0.636634	−	0.771166i	$-0.280326\pi$
$744$		0			0
$745$	10.4709	−	5.67075i	0.383625	−	0.207760i
$746$		0			0
$747$		−	10.3758i		−	0.379629i
$748$		0			0
$749$	−55.5350			−2.02921
$750$		0			0
$751$	−15.1306			−0.552125			−0.276062	−	0.961140i	$-0.589030\pi$
$751$	−15.1306			−0.552125			−0.276062	+	0.961140i	$0.589030\pi$
$752$		0			0
$753$		−	30.1154i		−	1.09747i
$754$		0			0
$755$	29.4413	−	15.9446i	1.07148	−	0.580282i
$756$		0			0
$757$			46.7505i			1.69918i	0.527447	+	0.849588i	$0.323149\pi$
$757$			46.7505i			1.69918i	−0.527447	+	0.849588i	$0.676851\pi$
$758$		0			0
$759$	−3.85204			−0.139820
$760$		0			0
$761$	27.2113			0.986410			0.493205	−	0.869913i	$-0.335825\pi$
$761$	27.2113			0.986410			0.493205	+	0.869913i	$0.335825\pi$
$762$		0			0
$763$		−	15.9202i		−	0.576350i
$764$		0			0
$765$	−8.40322	−	15.5164i	−0.303819	−	0.560996i
$766$		0			0
$767$			31.3573i			1.13224i
$768$		0			0
$769$	−8.97578			−0.323675			−0.161838	−	0.986817i	$-0.551742\pi$
$769$	−8.97578			−0.323675			−0.161838	+	0.986817i	$0.551742\pi$
$770$		0			0
$771$	−26.5811			−0.957295
$772$		0			0
$773$			29.6344i			1.06587i	0.846155	+	0.532937i	$0.178912\pi$
$773$			29.6344i			1.06587i	−0.846155	+	0.532937i	$0.821088\pi$
$774$		0			0
$775$	15.4651	−	23.7029i	0.555521	−	0.851433i
$776$		0			0
$777$			48.5776i			1.74271i
$778$		0			0
$779$	5.75100			0.206051
$780$		0			0
$781$	−25.7056			−0.919819
$782$		0			0
$783$			5.93796i			0.212205i
$784$		0			0
$785$	−24.5126	−	45.2620i	−0.874892	−	1.61547i
$786$		0			0
$787$		−	16.6729i		−	0.594325i	−0.954827	−	0.297162i	$-0.903960\pi$
$787$		−	16.6729i		−	0.594325i	0.954827	−	0.297162i	$-0.0960403\pi$
$788$		0			0
$789$	−10.5190			−0.374486
$790$		0			0
$791$	−34.2812			−1.21890
$792$		0			0
$793$			20.2436i			0.718872i
$794$		0			0
$795$	8.60169	−	4.65843i	0.305070	−	0.165217i
$796$		0			0
$797$		−	4.25284i		−	0.150643i	−0.997159	−	0.0753216i	$-0.976002\pi$
$797$		−	4.25284i		−	0.150643i	0.997159	−	0.0753216i	$-0.0239983\pi$
$798$		0			0
$799$	−1.79815			−0.0636141
$800$		0			0
$801$	−2.71428			−0.0959043
$802$		0			0
$803$		−	5.52903i		−	0.195115i
$804$		0			0
$805$	8.87784	−	4.80798i	0.312903	−	0.169459i
$806$		0			0
$807$			0.906058i			0.0318948i
$808$		0			0
$809$	−15.7772			−0.554696			−0.277348	−	0.960770i	$-0.589455\pi$
$809$	−15.7772			−0.554696			−0.277348	+	0.960770i	$0.589455\pi$
$810$		0			0
$811$	−39.9011			−1.40112			−0.700559	−	0.713594i	$-0.747066\pi$
$811$	−39.9011			−1.40112			−0.700559	+	0.713594i	$0.747066\pi$
$812$		0			0
$813$			11.0749i			0.388412i
$814$		0			0
$815$	−20.4100	−	37.6866i	−0.714930	−	1.32010i
$816$		0			0
$817$			29.1890i			1.02119i
$818$		0			0
$819$	−19.8198			−0.692558
$820$		0			0
$821$	0.842179			0.0293923			0.0146961	−	0.999892i	$-0.495322\pi$
$821$	0.842179			0.0293923			0.0146961	+	0.999892i	$0.495322\pi$
$822$		0			0
$823$			51.5407i			1.79660i	0.439388	+	0.898298i	$0.355195\pi$
$823$			51.5407i			1.79660i	−0.439388	+	0.898298i	$0.644805\pi$
$824$		0			0
$825$	15.4415	+	10.0749i	0.537604	+	0.350762i
$826$		0			0
$827$			0.433745i			0.0150828i	0.999972	+	0.00754140i	$0.00240052\pi$
$827$			0.433745i			0.0150828i	−0.999972	+	0.00754140i	$0.997599\pi$
$828$		0			0
$829$	−24.8652			−0.863604			−0.431802	−	0.901968i	$-0.642122\pi$
$829$	−24.8652			−0.863604			−0.431802	+	0.901968i	$0.642122\pi$
$830$		0			0
$831$	−3.74825			−0.130025
$832$		0			0
$833$		−	92.1897i		−	3.19418i
$834$		0			0
$835$	−15.6695	−	28.9334i	−0.542265	−	1.00128i
$836$		0			0
$837$		−	5.66037i		−	0.195651i
$838$		0			0
$839$	−43.0915			−1.48768			−0.743842	−	0.668355i	$-0.766999\pi$
$839$	−43.0915			−1.48768			−0.743842	+	0.668355i	$0.766999\pi$
$840$		0			0
$841$	6.25933			0.215839
$842$		0			0
$843$		−	17.7546i		−	0.611501i
$844$		0			0
$845$	−15.7820	+	8.54706i	−0.542917	+	0.294028i
$846$		0			0
$847$		−	11.2282i		−	0.385804i
$848$		0			0
$849$	−19.6755			−0.675260
$850$		0			0
$851$	−11.7403			−0.402452
$852$		0			0
$853$		−	45.0330i		−	1.54190i	−0.636895	−	0.770951i	$-0.719781\pi$
$853$		−	45.0330i		−	1.54190i	0.636895	−	0.770951i	$-0.280219\pi$
$854$		0			0
$855$	−8.30873	+	4.49977i	−0.284153	+	0.153889i
$856$		0			0
$857$			45.9359i			1.56914i	0.620039	+	0.784571i	$0.287117\pi$
$857$			45.9359i			1.56914i	−0.620039	+	0.784571i	$0.712883\pi$
$858$		0			0
$859$	47.9577			1.63629			0.818147	−	0.575009i	$-0.195002\pi$
$859$	47.9577			1.63629			0.818147	+	0.575009i	$0.195002\pi$
$860$		0			0
$861$	5.88246			0.200474
$862$		0			0
$863$			47.8219i			1.62788i	0.580952	+	0.813938i	$0.302681\pi$
$863$			47.8219i			1.62788i	−0.580952	+	0.813938i	$0.697319\pi$
$864$		0			0
$865$	15.7498	+	29.0818i	0.535510	+	0.988809i
$866$		0			0
$867$		−	45.2744i		−	1.53760i
$868$		0			0
$869$	−19.0455			−0.646075
$870$		0			0
$871$	−4.58547			−0.155373
$872$		0			0
$873$			8.15818i			0.276113i
$874$		0			0
$875$	−48.1634	−	3.94615i	−1.62822	−	0.133404i
$876$		0			0
$877$		−	56.1039i		−	1.89449i	−0.320505	−	0.947247i	$-0.603853\pi$
$877$		−	56.1039i		−	1.89449i	0.320505	−	0.947247i	$-0.396147\pi$
$878$		0			0
$879$	16.3514			0.551519
$880$		0			0
$881$	27.8879			0.939567			0.469784	−	0.882782i	$-0.344332\pi$
$881$	27.8879			0.939567			0.469784	+	0.882782i	$0.344332\pi$
$882$		0			0
$883$		−	48.5203i		−	1.63284i	−0.577459	−	0.816419i	$-0.695956\pi$
$883$		−	48.5203i		−	1.63284i	0.577459	−	0.816419i	$-0.304044\pi$
$884$		0			0
$885$	7.28191	+	13.4459i	0.244779	+	0.451979i
$886$		0			0
$887$			42.3930i			1.42342i	0.702475	+	0.711708i	$0.252078\pi$
$887$			42.3930i			1.42342i	−0.702475	+	0.711708i	$0.747922\pi$
$888$		0			0
$889$	−27.3364			−0.916833
$890$		0			0
$891$	3.68751			0.123536
$892$		0			0
$893$			0.962878i			0.0322215i
$894$		0			0
$895$	−32.9829	+	17.8626i	−1.10250	+	0.597081i
$896$		0			0
$897$		−	4.79006i		−	0.159935i
$898$		0			0
$899$	−33.6111			−1.12099
$900$		0			0
$901$	34.5226			1.15011
$902$		0			0
$903$			29.8562i			0.993553i
$904$		0			0
$905$	37.1081	−	20.0967i	1.23352	−	0.668036i
$906$		0			0
$907$			27.2037i			0.903283i	0.892200	+	0.451641i	$0.149161\pi$
$907$			27.2037i			0.903283i	−0.892200	+	0.451641i	$0.850839\pi$
$908$		0			0
$909$	−2.75538			−0.0913903
$910$		0			0
$911$	38.6906			1.28188			0.640938	−	0.767592i	$-0.278545\pi$
$911$	38.6906			1.28188			0.640938	+	0.767592i	$0.278545\pi$
$912$		0			0
$913$			38.2607i			1.26625i
$914$		0			0
$915$	4.70106	+	8.68042i	0.155412	+	0.286966i
$916$		0			0
$917$		−	38.0946i		−	1.25800i
$918$		0			0
$919$	38.3399			1.26472			0.632359	−	0.774676i	$-0.282087\pi$
$919$	38.3399			1.26472			0.632359	+	0.774676i	$0.282087\pi$
$920$		0			0
$921$	0.589802			0.0194347
$922$		0			0
$923$		−	31.9652i		−	1.05215i
$924$		0			0
$925$	47.0628	+	30.7063i	1.54742	+	1.00962i
$926$		0			0
$927$		−	11.5307i		−	0.378718i
$928$		0			0
$929$	−39.2640			−1.28821			−0.644105	−	0.764937i	$-0.722770\pi$
$929$	−39.2640			−1.28821			−0.644105	+	0.764937i	$0.722770\pi$
$930$		0			0
$931$	−49.3659			−1.61790
$932$		0			0
$933$			13.3850i			0.438206i
$934$		0			0
$935$	30.9870	+	57.2168i	1.01338	+	1.87119i
$936$		0			0
$937$		−	18.8708i		−	0.616483i	−0.951308	−	0.308242i	$-0.900259\pi$
$937$		−	18.8708i		−	0.616483i	0.951308	−	0.308242i	$-0.0997405\pi$
$938$		0			0
$939$	15.9356			0.520040
$940$		0			0
$941$	−33.8851			−1.10462			−0.552311	−	0.833638i	$-0.686254\pi$
$941$	−33.8851			−1.10462			−0.552311	+	0.833638i	$0.686254\pi$
$942$		0			0
$943$			1.42168i			0.0462962i
$944$		0			0
$945$	−8.49866	+	4.60263i	−0.276461	+	0.149723i
$946$		0			0
$947$		−	24.6489i		−	0.800980i	−0.916301	−	0.400490i	$-0.868840\pi$
$947$		−	24.6489i		−	0.800980i	0.916301	−	0.400490i	$-0.131160\pi$
$948$		0			0
$949$	6.87543			0.223186
$950$		0			0
$951$	16.4627			0.533841
$952$		0			0
$953$			44.0917i			1.42827i	0.700007	+	0.714136i	$0.253180\pi$
$953$			44.0917i			1.42827i	−0.700007	+	0.714136i	$0.746820\pi$
$954$		0			0
$955$	8.34830	−	4.52120i	0.270145	−	0.146303i
$956$		0			0
$957$		−	21.8963i		−	0.707806i
$958$		0			0
$959$	−35.1475			−1.13497
$960$		0			0
$961$	1.03982			0.0335426
$962$		0			0
$963$			12.8485i			0.414037i
$964$		0			0
$965$	−18.9733	−	35.0339i	−0.610773	−	1.12778i
$966$		0			0
$967$		−	9.29052i		−	0.298763i	−0.988780	−	0.149381i	$-0.952272\pi$
$967$		−	9.29052i		−	0.298763i	0.988780	−	0.149381i	$-0.0477282\pi$
$968$		0			0
$969$	−33.3468			−1.07125
$970$		0			0
$971$	21.6614			0.695147			0.347574	−	0.937653i	$-0.387006\pi$
$971$	21.6614			0.695147			0.347574	+	0.937653i	$0.387006\pi$
$972$		0			0
$973$		−	18.8096i		−	0.603008i
$974$		0			0
$975$	−12.5282	+	19.2017i	−0.401225	+	0.614947i
$976$		0			0
$977$			60.2392i			1.92722i	0.267308	+	0.963611i	$0.413866\pi$
$977$			60.2392i			1.92722i	−0.267308	+	0.963611i	$0.586134\pi$
$978$		0			0
$979$	10.0089			0.319887
$980$		0			0
$981$	−3.68327			−0.117598
$982$		0			0
$983$			41.4407i			1.32175i	0.750495	+	0.660877i	$0.229815\pi$
$983$			41.4407i			1.32175i	−0.750495	+	0.660877i	$0.770185\pi$
$984$		0			0
$985$	8.30086	+	15.3274i	0.264487	+	0.488370i
$986$		0			0
$987$			0.984887i			0.0313493i
$988$		0			0
$989$	−7.21568			−0.229445
$990$		0			0
$991$	18.3043			0.581454			0.290727	−	0.956806i	$-0.406103\pi$
$991$	18.3043			0.581454			0.290727	+	0.956806i	$0.406103\pi$
$992$		0			0
$993$			32.0995i			1.01865i
$994$		0			0
$995$	2.00293	−	1.08473i	0.0634972	−	0.0343882i
$996$		0			0
$997$			18.1938i			0.576204i	0.957600	+	0.288102i	$0.0930242\pi$
$997$			18.1938i			0.576204i	−0.957600	+	0.288102i	$0.906976\pi$
$998$		0			0
$999$	11.2388			0.355581

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.c.1609.35	yes	38
5.4	even	2	inner	4020.2.g.c.1609.16	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.c.1609.16	✓	38	5.4	even	2	inner
4020.2.g.c.1609.35	yes	38	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(1.96624 - 1.06486i) q^{5} -4.32230i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(1.96624 - 1.06486i) q^{5} -4.32230i q^{7} -1.00000 q^{9} +3.68751 q^{11} +4.58547i q^{13} +(1.06486 + 1.96624i) q^{15} +7.89141i q^{17} +4.22571 q^{19} +4.32230 q^{21} +1.04462i q^{23} +(2.73216 - 4.18752i) q^{25} -1.00000i q^{27} -5.93796 q^{29} +5.66037 q^{31} +3.68751i q^{33} +(-4.60263 - 8.49866i) q^{35} +11.2388i q^{37} -4.58547 q^{39} +1.36096 q^{41} +6.90749i q^{43} +(-1.96624 + 1.06486i) q^{45} +0.227862i q^{47} -11.6823 q^{49} -7.89141 q^{51} -4.37470i q^{53} +(7.25051 - 3.92667i) q^{55} +4.22571i q^{57} +6.83840 q^{59} +4.41474 q^{61} +4.32230i q^{63} +(4.88286 + 9.01611i) q^{65} +1.00000i q^{67} -1.04462 q^{69} -6.97099 q^{71} -1.49940i q^{73} +(4.18752 + 2.73216i) q^{75} -15.9385i q^{77} -5.16488 q^{79} +1.00000 q^{81} +10.3758i q^{83} +(8.40322 + 15.5164i) q^{85} -5.93796i q^{87} +2.71428 q^{89} +19.8198 q^{91} +5.66037i q^{93} +(8.30873 - 4.49977i) q^{95} -8.15818i q^{97} -3.68751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) \) 38 * q - 2 * q^5 - 38 * q^9	\( 38 q - 2 q^{5} - 38 q^{9} + 24 q^{11} + 2 q^{15} - 16 q^{19} + 12 q^{21} + 4 q^{25} - 56 q^{29} - 4 q^{31} - 2 q^{35} + 60 q^{41} + 2 q^{45} - 70 q^{49} + 12 q^{55} - 52 q^{59} + 48 q^{61} + 16 q^{65} - 12 q^{69} + 12 q^{75} - 24 q^{79} + 38 q^{81} + 16 q^{85} - 76 q^{89} + 44 q^{91} + 36 q^{95} - 24 q^{99}+O(q^{100}) \) 38 * q - 2 * q^5 - 38 * q^9 + 24 * q^11 + 2 * q^15 - 16 * q^19 + 12 * q^21 + 4 * q^25 - 56 * q^29 - 4 * q^31 - 2 * q^35 + 60 * q^41 + 2 * q^45 - 70 * q^49 + 12 * q^55 - 52 * q^59 + 48 * q^61 + 16 * q^65 - 12 * q^69 + 12 * q^75 - 24 * q^79 + 38 * q^81 + 16 * q^85 - 76 * q^89 + 44 * q^91 + 36 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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